As Ben Green points out above, it is too much to ask for quadratic behaviour. For example, consider the subset of that consists of all points , where is a generator of the multiplicative group mod (which I have denoted by ). Then if , then we have such that , which gives us that , and therefore and therefore . This example can be embedded into the integers in the usual way, and shows that it is possible for an example to be non-quadratic in a fairly fundamental way. So a good conjecture would somehow have to come up with a definition that could unify this example with the quadratic one. It seems quite a good idea to focus on graphs of functions: I think it may be possible to convert any example into a graph by a process such as multiplying by a random number mod , for some prime close to , splitting up into intervals of length roughly , and throwing away a certain proportion of the set so that each element lives in the first half of its interval and there is only one such element per interval.

]]>I fisrt took from and wrote the condition , I think it was a better precaution, we just say that is small when is close to instead of previous “ is small when “is small”.

]]>Maybe a conjecture could be of the form :

For any Sidon set , with , such that there exists , such that ,

Where

_ is a fonction that we have to find , that is close to identity

_ is a set of Sidon sets with some quadratic property that we have to precise

_ is a increasing fonction from to , where is order relations that we also have to precise (for example we can take iff )

_ is such that is small when is small.

The idea is that if we get close to we can find a Sidon set with a quadratic structure close to . But maybe it is not enough and you want to have itself the "quadratic property", not just close to a set that has it….

We can also state a "skeleton" of direct conjecture – that could maybe help to find/solve the inverse problem that we try to formulate – as :

For any , there exists such that

where is close to when is small (for )

As I'm not familiar with the problem, I won't take the risk to give explicit , , so I don't really "say something" neither. But just a direction, so that we might be able to discuss on the choice of each parameter.

(This comment is identical to that I posted yesterday, but with a proper formatting attempted.)

I wonder whether one can use Ben’s idea in conjunction with Ruzsa’s construction, as follows. Let be a prime, and consider indices (discrete logarithms) with respect to a fixed primitive root modulo . With every integer associate the residue class of modulo . The set of all resulting elements of is a Sidon set. Define by . Writing , we see that to answer Erdos’ question, it suffices to show that the full image misses some, say, consecutive elements of . A simple heuristic shows that is most certainly true (a typical element of is missing in with probability about ), but is there a hope to prove this? Notice also that we do not need to have the result for all pairs (where is a primitive root modulo ), but just for any infinite sequence of such pairs.

]]>I wonder whether one can use Ben’s idea in conjunction with Ruzsa’s

construction, as follows. Let $p$ be a prime, and consider indices (discrete

logarithms) with respect to a fixed primitive root modulo $p$. With every

integer $x\in[1,p-1]$ associate the residue class of $(p-1)x+p\,{\rm

ind}\,(x)$ modulo $p(p-1)$. The set of all $p-1$ resulting elements of

${\mathbb Z}_{p(p-1)}$ is a Sidon set. Define $\varphi\colon

[1,p-1]\to{\mathbb Z}_{p-1}$ by $\varphi(x)=x+{\rm ind}\, x\pmod {p-1}$.

Writing $(p-1)x+p\,{\rm ind}\,(x)=(x+{\rm ind}\,(x))p -x$, we see that to

answer Erdos’ question, it suffices to show that the full image

$\varphi([1,p-1])$ misses some, say, $\log\log\log p$ consecutive elements of

${\mathbb Z}_{p-1}$. A simple heuristic shows that is most certainly true (a

typical element of ${\mathbb Z}_{p-1}$ is missing in $\varphi([1,p-1])$ with

probability about $1/e$), but is there a hope to \emph{prove} this? Notice

also that we do not need to have the result for all pairs $(p,g)$ (where $g$

is a primitive root modulo $p$), but just for any infinite sequence of such

pairs.

my name is Yan， I read “what is implied by “implies””you wrote

And I have a question for you. Can you just spend a little bit time and give me some help？

it is annoying question about vacuous proof.

for example， let’s prove ∅ is the subset of every set A.

This statement can be translated into another way：for every x， if x is the element of ∅，then x is the element of A.

since x is the element of ∅ is always false. So this statement is vacuous true.

My problem is why can‘t we substitue x by non-exist things，for example， what if x represents unicorn？then unicorn is an element of ∅ will be true not false？so x is the element of empty set is not always true

by this problem， can we get the conclusion that all the varibles in the statement can just be substituted by exist things? which means the scope of a variable can just be exist thing?

I appreciate your help Dr Gowers~

]]>As a point of history, the Erdos-Turan proof and the Lindstrom proof both give (iirc, the “+1” can be improved to “+1/2”), although E-T only noted .

]]>Let be a dynamical system, and suppose that is ergodic, and that . Let be an arbitrary set (measurable, of course), and let be a randomly chosen point of . Now let be the set of integers where is parameter to be named later. Since is ergodic, the expected size of is ; perhaps we can take so that . I claim that is pressured to be a Sidon set, at least after removing a small number of elements.

Why? Suppose that are all in , and , contrary to the claimed Sidon-ness of (set ). This means that the two points , are both in , and so are “close together”. But also maps these two close points to , , which are not only close to each other but close to the original two points. My intuition is that if is sufficiently strongly mixing, then it should be very rare (i.e., very few ) that a small power of maps two points in to two points in . Caveat: I don’t know what “sufficiently strongly mixing”, “very rare”, or “small power” actually need to mean to make this work, if indeed there is any way to make it work.

So here’s the promised example, which is really a Bose-Chowla set in disguise. Let , Lebesgue measure, and define to be . This is a linear map with determinant 6, so it is ergodic. Set , and let be any of the sets with (or replace the special role played by with , that works, too). Let . (Above, I suggested taking random and fixed , but here I'm taking one and many 's. Oh well.) All 32 of the resulting sets are Sidon sets!

All of the constructions of Sidon sets that give are based on finite fields, and the error term in the maximum possible size of a Sidon set are from the gaps between prime numbers. Any construction that gives without using primes would be interesting, and possibly the first improvement since the early 1940s.

]]>FYI, the constructions of Bose & Chowla (mod ), Singer (mod ), and Ruzsa (mod ) are given in my now old survey/bibliography at Electronic Journal of Combinatorics (http://www.combinatorics.org/ojs/index.php/eljc/article/view/ds11).

]]>Indeed, it seems very hard to improve Lindstrom’s result in any way. Erdos conjecture is surely false, as most likely one can dilate a Sidon subset of Z/qZ of size q^{1/2} so as to have a gap of size q^{1/2}logloglog q (say), and then “unwrap” so as to get a Sidon set of size q^{1/2} in an interval of length appreciably smaller than q. However, I have no idea how to achieve this with any of the known constructions of large Sidon sets modulo q. Being able to find such large gaps is most likely a generic property (i.e. has nothing to do with being Sidon) but I have no idea how to prove that either.

]]>Mark, this is an interesting idea, and I like this conjecture of Helfgott and Venkatesh very much. However I suspect that *very* dense Sidon sets, of size really close to the maximum of $\sqrt{N}$, are rather well-distributed in residue classes to small moduli.

See

Ah, I see! But the discussion at that point was about infinite Sidon sets, so I was distracted. Thanks, Benoît.

]]>What Tim Gowers meant is that in a given interval [0,N], there are far more logs of prime than integer.

]]>IErdos offered $500 for a proof of an upper bound of $n^{1/2} + O(1)$. ]]>

Actually, what I wrote in 2) isn’t properly correct as it stands, but I see a similar problem with the example in the post. Namely, how can one guarantee that the set actually has size at least $m/\log m$? For many different $p\leq m$ are such that the integer part of $\log p$ are the same. Surely one needs the primes to be spaced at least some multiplicative constant apart, whence one only gets a set of size around $\log m$?

]]>Looking at the data for n=44 the size distribution for all Sidon sets is unimodal, the most common size is 7 and the maximum is 9. Here is the list of {size, number of Sidon sets} pairs for n=44.

{{0, 1}, {1, 44}, {2, 946}, {3, 13 244}, {4, 129 360}, {5, 845 408},

{6, 3 157 104}, {7, 4 748 144}, {8, 1 462 854}, {9, 27 202}}

For larger n I only had data for restricted Sidon set, the ones which are well distributed mod p, but they show a similar distribution.

Again this is for very(!) small n, but it looks like as “close” to maximum size one might need to be close by something additive rather than a factor.

]]>One of my old programs looked for Sidon sets such that every initial segment is as balanced as possible mod p, no residue classes differing by more than 1.

There are of course fewer such sets but their sizes seem to grow at a rate comparable with that of general Sidon sets, perhaps smaller by some p-dependent factor. The same looks plausible when one uses several prime at the same time.

Here are two examples which are balanced mod 2, 3, and 5

{1, 8, 15, 34, 42, 53, 65, 96, 109, 112, 113, 114}

{1, 8, 15, 24, 47, 64, 66, 77, 85, 88, 113, 114}

I took a look at some old data on Sidon sets which I had on my laptop, I once had a long wait in an airport which led to some simple computer experiments with Sidon sets, and from those data it looks like c is 4.1… in the estimate for the number of Sidon sets.

However this is data from what my laptop could do at the time so they are only for n up to 44, for general Sidon sets.

]]>*Thanks — corrected now.*

2) You can also construct quite nasty examples from iterating , and the floor functions. For example, you can start from the primes and instead of I think one obtains a Sidon set of roughly the same density by considering . You could also take where is taken from a Sidon set of a completely different nature. I imagine one could repeat this process. It’s difficult to guess what kind of structure is preserved by repeated logarithms and floor functions…

]]>